integral of (9x+6)/((x-6)(x+4))

Lösung

\int \frac{9 x + 6}{( x - 6 ) ( x + 4 )} d x

6 ln ∣ x - 6 ∣ + 3 ln ∣ x + 4 ∣ + C

Schritte zur Lösung

\int \frac{9 x + 6}{( x - 6 ) ( x + 4 )} d x

Multipliziere aus

\frac{9 x + 6}{( x - 6 ) ( x + 4 )} : \frac{9 x}{( x - 6 ) ( x + 4 )} + \frac{6}{( x - 6 ) ( x + 4 )}

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{9 x}{( x - 6 ) ( x + 4 )} d x + \int \frac{6}{( x - 6 ) ( x + 4 )} d x

\int \frac{9 x}{( x - 6 ) ( x + 4 )} d x = 9 (\frac{3}{5} ln ∣ x - 6 ∣ + \frac{2}{5} ln ∣ x + 4 ∣)

\int \frac{6}{( x - 6 ) ( x + 4 )} d x = 6 (\frac{1}{10} ln ∣ x - 6 ∣ - \frac{1}{10} ln ∣ x + 4 ∣)

= 9 (\frac{3}{5} ln ∣ x - 6 ∣ + \frac{2}{5} ln ∣ x + 4 ∣) + 6 (\frac{1}{10} ln ∣ x - 6 ∣ - \frac{1}{10} ln ∣ x + 4 ∣)

Vereinfache

9 (\frac{3}{5} ln ∣ x - 6 ∣ + \frac{2}{5} ln ∣ x + 4 ∣) + 6 (\frac{1}{10} ln ∣ x - 6 ∣ - \frac{1}{10} ln ∣ x + 4 ∣) : 6 ln ∣ x - 6 ∣ + 3 ln ∣ x + 4 ∣

= 6 ln ∣ x - 6 ∣ + 3 ln ∣ x + 4 ∣

Füge eine Konstante zur Lösung hinzu

= 6 ln ∣ x - 6 ∣ + 3 ln ∣ x + 4 ∣ + C

\int_{1}^{2} \frac{x + 1}{x ( x ^{2} + 1 )} d x

d er i v a t i v e y = ln (x^{2} + 4)

x \to 6 lim (x)

\int_{0}^{3 π^{2}} θ cos^{2} (θ^{\frac{3}{2}}) d θ

\frac{\partial}{\partial x} (\frac{2}{x} - x)